One would argue that nature is amoral since it does not really have respect for life. Meaning, it causes life but it also causes death, suffering, destruction, misery etc.
And to derive something you perceive as favorable from nature as a moral code while neglecting what you do not desire (destruction, for example) is contrary to reason.

The short answer to your question is, as I understand Objectivisim: A resounding No.

As to the sub-parts to you post:

I'd agree that nature is amoral. Not because it has no respect for life but because morality applies only to man. Morality is a code of values accepted by choice.
To quote from OPAR (chapter 7 "The Good"):

Plants and animals pursue values, but
not moral values; they have goals, but
not ethics. Moral values are a
subcategory of values, defined by two
conditions. "Moral Values" are
chosen values of a fundamental nature.

As for deriving a moral code from observing nature; "You're doing it wrong" as the kids say. Whereas metaphysics and epistemology are factual subjects, morality is evaluative. To quote again from OPAR (Ch 7, "The Good"):

Its [ethics/morality] concern is not
only to describe, but to prescribe for
man. Ethics is the branch philosophy
that, in Ayn Rand's words, provides "a
code of values to guide man's choices
and actions - the choice and actions
that determine the purpose and the
course of his life".

According to Objectivisim, such a code
mus deal with three basic,
interrelated questions. For what end
should a man live? By what fundamental
principle should he act in order to
achieve this end? Who should profit
from his actions?

Rand answers are: life, rationality and oneself. While her arguments for each of these is much too long for this post non of them appear (to me) to be based on simple/direct observation of nature.

If you want to live, then it is a fact that you must act in certain ways and not in others. This is the base of an objective morality. It is true that the choice to live is pre-moral. However, morality can only apply to those who have chosen life. Morality is a code of values to guide one's actions. Only those who have chosen life have values and take actions. Morality is meaningless in any other context.

Some people are bothered by the fact that you do not have to choose to live---to them morality must completely constrain every choice a person makes, and it is simply unacceptable for there to be a choice that morality does not have jurisdiction over. Morality must be universal, they argue. To those who argue this, I ask: why? What facts of reality have given rise to the requirements on morality that you insist must be met?

For those who chose life, morality is objective and universally applicable. For those who don't, morality simply cannot apply to them at all.

Some people get distracted by a tendency to look only at a single "tree" instead of a collection of "trees," i.e., a "forest." If one looks at humans as a class, one finds that they all have the capacity to initiate life-sustaining actions or not, which is equivalent to choosing to live or not. Nearly all humans are, in fact, life-seekers (mostly), not intentional death-choosers. If or when any humans decide to choose death (other than in a context in which the only alternative is some unbearable existence in chronic suffering and misery), they are perfectly capable of coexisting (while they die) with all the vast numbers of life-seekers. They are free to die as they wish, with no need or reason to seek anything from life-seekers or to attack life-seekers in any manner. Hypothetical death-choosers simply pose no moral problem at all for life-seekers. Life-seekers don't need anything from death-choosers and have no reason (nothing to gain from them) to pay any attention to them at all (unless a life-seeker thinks he might be able to help a death-chooser to start seeking life again, and the life-seeker wants to try to help).

Another metaphor is a box full of "white marbles" (life-seekers) and short-lived, self-extinguishing "black marbles" (death-choosers). One may find occasional black marbles among all the white ones, but so what? There is no issue for the white marbles in that observation (other than possible self-defense if attacked). Morality applies to the white marbles. The black marbles don't need morality; no specific course of action is needed to achieve death, other than to refrain from initiating any life-sustaining actions. One does not need morality to tell one how to die.

As another answer has already pointed out, Objectivist morality builds on the causal relation between living and life-seeking actions, and that causal relation is a fact of reality. This is, indeed, a revolutionary discovery and approach in the history of philosophy.

Does Objectivism claim that we can derive a moral code on something that is 'amoral'?

Well, yes, I suppose reality is amoral, and we can derive a moral code from reality.

One would argue that nature is amoral since it does not really have respect for life. Meaning, it causes life but it also causes death, suffering, destruction, misery etc.

I agree that nature is amoral, but it doesn't cause life and death. That's like saying physics causes gravity, or reality causes existence. Nature is simply what is; it has no choice, no volition. Nature IS life and death.

Morality is the chosen. It only applies to living beings with free choice -- meaning humans.

And to derive something you perceive as favorable from nature as a moral code while neglecting what you do not desire (destruction, for example) is contrary to reason.

A proper morality most definitely takes into account what we do not desire. Morality guides us toward life and happiness, and away from death. The idea is to conform with reality and nature, not to fight them.